• R.Parimalazhagan is currently working as Professor and Head , Depart-

ment of Science and Humanities in Karpagam College of Engineering ,

Definition 2.1[13] : A subset A of a topological space (X, r) is called generalized closed set (briefly g-closed) if cl(A)� U,whenever A � U and U is open in X .Definition 2.2 : A subset A of a topological space (X, r) iscalled * closed set. If cl(int(A))� U, whenever A � U and Uis g-open in X .Definition 2.3[25] : A subset A of a topological space (X, r) is called g*-closed if cl(A)� U,whenever A � U and U is g-openin X .Definition 2.4: A map f : (X, r) �(Y, a) from a topologicalspace X into a topological space Y is called g-continuous iff-1 (V) is g-closed in X for every closed set V of Y.Definition 2.5: A map f : (X, r) �(Y, a) from a topologicalspace X into a topological space Y is called *-continuous iff-1 (V) is *- closed in X for every closed set V of Y.Definition 2.6: A map f : (X, r) �(Y, a) from a topologicalspace X into a topological space Y is called irresolute if f-1 (V)is semi-closed in X for every semi-closed set V of Y.Definition 2.7[5]: A map f : (X, r) �(Y, a) from a topologicalspace X into a topological space Y is called semi-generalizedcontinuous(briefly sg continuous) if f-1 (V) is sg-closedin X for every closed set V of Y.

3. J3*- Continuous Maps

In this section we introduce the concept of *-Continuous maps in topological spaces.

Definition 3.1 Let f : X � Y from a topological space X into a

topological space Y is called *-continuous if the inverse im- age of every closed set in Y is *-closed in X.

Theorem 3.2 If a map f : X � Y from a topological space X into

a topological space Y is continuous, then it is *-continuous but not conversely.

Proof: Let f : X � Y be continuous. Let F be any closed set in

Y. Then the inverse image f -1 (F) is closed in X. Since every closed set is *-closed, f -1 (F) is *-closed in X. Therefore f is

*-continuous.

Remark 3.3 The converse of the theorem 3.2 need not be true as seen from the following example

Theorem 3.11: If a map f : X � Y from a topological space X

into a topological space Y is gs-continuous, then it is * - continuous but not conversely.

Proof: Let f : X � Y be sg-continuous. Let V be any gs-closed

(a) For each point x E X and each open set V in Y with f ( x ) EV,there is a * -open set U in X such that x E U , f (U ) c V .(b) For every subset A of X , f ( * (A)) ccl(int (f(A))) holds.(c) For each subset B of Y, * (f-1(B)) c f-1(cl(int ((A)). Proof: (i) Assume that f : X � Y be *-continuous. Let G be open in Y.Then G c is closed in Y . Since f is *-continuous ,f-1 (G c ) is *-closed in X. But f-1 (Gc )= X - f-1 (G) .Thus X -f-1 (G) is *- closed in X and so f-1 (G) is *-open in X. There-fore(a) implies (b).Conversely assume that the inverse image of each open set inY is *-open in X. Let F be any closed set in Y. The Fc is openin Y. By assumption, f-1 (Fc ) is *-open in X. But f-1 (Fc ) = X

is not *-continuous since for a closed set {a,c} in Yf-1 ({a, c}) = {a, b} is not *-closed in X .

Theorem 3.16 If f : X � Y and g :Y � Z be any two functions,

Then g o f: X � z is *-continuous if g is continuous and f is*-continuousProof: Let V be a closed set in Z, Since g is continuous, g-1(V) is closed in V. and since f is * continuous.f-1(g-1(V) is *-closedin X . thus g@ f is * continuous.

4 J3 *- Irresolute Maps

Definition 4.1 A map f : X � Y from a topological space Y is

called *-irresolute if the inverse image of every *-closed set in Y is *-closed in X.

Theorem 4.2 A map f : X � Y is * -irresolute if and only if

the inverse image of every * -open set in Y is * -open in X. Proof: Assume that f is *-irresolute. Let A be any *-open set in Y. Then Ac is *-closed set in Y. Since f is *-irresolute,

f-1(Ac) is *-closed in X.But f-1(Ac) = X -f-1(A) and so f-1(A) is * -

open in X. Hence the inverse image of every *-open set in Y is*-open in X. Conversely assume that the inverse image of every * -open set in Y is *-open in X. Let A be any *-closed set in Y. Then Ac is *-open in Y. By assumption, f-1(Ac) is *-open in X. But f-1(Ac) = X - f-1(A) and so f-1(A) is*-closed in X. Therefore f is *-irresolute.

Theorem 4.3 If a map f : X � Y is *-irresolute, then it is *- continuous but not conversely.

Proof: Assume that f is *-irresolute. Let F be any closed set in

Y. Since every closed set is *-closed, F is *-closed in Y. Since f is *-irresolute, f-1(F) is *-closed in X. Therefore f is *-conti nuous.

Theorem 4.6 Let X, Y and Z be any topological spaces. For any

Proof: Let F be any *-closed set in Z. Since g is *- continuous

, g-1(F) is *-closed in Y. Since f is *-irresolute, f-1(g-1(F)) is *-

5 Pasting Lemma for J3 * -Continuous Maps

Theorem 5.1 Let X = A u B be a topological space with topolo-

gy " and Y be a topological space with topology a.Let f : (A, "/A) � (Y, a) and g :(B, "/B) � (Y, a) be *- conti-nuous maps such that f(x) = g(x) for every x @A n B. Suppose that A and B are *-closed sets in X. Then the combination

a : (X, ") � (Y, a) is *-continuos.

Proof: Let F be any closed set in Y. Clearly a-1(F) = f-1(F) u

g-1(F)= C u D where C = f-1(F) and D = g-1(F). But C is *-closed

in A and A is *-closed in X and so C is *-closed in X. Since we have proved that if B@A@ X, B is *-closed in A and A is

*-closed in X then B is *-closed in X. Also C u D is *-closedin X. Therefore a-1(F) is *-closed in X. Hence a is *- continous.